[FOM] Characterization of the real numbers

William Tait wwtx at earthlink.net
Mon Feb 7 15:30:11 EST 2005

X (a complete ordered set) having a countable dense subset means that 
there is a function F: X -> X and a z in X such that the orbit of F 
starting at z (closure of {z} under F) is dense in X.

Best,  Bill Tait

On Feb 6, 2005, at 7:10 PM, JoeShipman at aol.com wrote:

> Yes, but "having a countable dense subset" involves significantly 
> higher logical complexity, you have to define "countable subset".  I'd 
> like to avoid defining the integers if possible.
> Mayberry wrote:
> How about "complete linear ordering without endpoints having a 
> countable
> dense subset". I think this is Cantor's characterization.
> --On 06 February 2005 15:23 -0500 JoeShipman at aol.com wrote:
>> What is the simplest characterization of the real numbers?  That is, 
>> what
>> is the simplest description of a structure, any model of which is
>> isomorphic to the real numbers?
>> A standard way of characterizing the real numbers is "ordered field 
>> with
>> the least upper bound property".  But do I need to refer to field
>> operations?  "Dense ordering without endpoints and the least upper 
>> bound
>> property" isn't sharp enough.  "Homogenous dense ordering with the 
>> least
>> upper bound property" looks better, except that it doesn't rule out 
>> the
>> "long line" (product of the set of countable ordinals with [0,1} in
>> dictionary order, with intial point removed).  (It also doesn't rule 
>> out
>> the reversed long line, or the symmetric long line.)
>> The best I can do without referring to relations other than the order
>> relation is "dense ordering with least upper bound property, 
>> isomorphic
>> to any of its nonempty open intervals". Can anyone improve on this?
>> -- JS
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